Domains with Non-Compact Automorphism Group: A Survey☆

We survey results arising from the study of domains in C^n with non-compact automorphism group. Beginning with a well-known characterization of the unit ball, we develop ideas toward a consideration of weakly pseudoconvex (and even non-pseudoconvex) domains with particular emphasis on characterizations of (i) smoothly bounded domains with non-compact automorphism group and (ii) the Levi geometry of boundary orbit accumulation points. 
Particular attention will be paid to results derived in the past ten years.

Domains with Non-Compact Automorphism Group: A Survey

https://www.jstage.jst.go.jp/article/math1924/15/2/15_2_385/_pdf

Automorphisms of bounded Reinhardt domains

The future tube in ℂ n +1 is the unbounded domain τ+ = z = (z o,...,z n ) ∈ ℂ n +1 : (Im z o)2 > (Imz 1)2 +...+ (Im z n )2 Im z o > 0. In other words, τ+ is a tube domain over the future cone V + = y ∈ ℝ n +1 : y 2 0 > y 2 1 +...+ y 2 n , y 0 > 0. The domain τ+ is biholomorphically equivalent to a classical Cartan domain of the IVth type, hence to a bounded symmetric domain in ℂ n +1. The future tube τ+ in ℂ4 (n = 3) is important in mathematical physics, especially in axiomatic quantum field theory, being the natural domain of definition of holomorphic relativistic fields. These specific features of the future tube motivated its investigation by mathematicians and physicists. Beginning with Elie Cartan’s classification of bounded symmetric domains, these domains were examined in many papers where the complex structure of their boundaries, integral representations, boundary values of holomorphic functions and so on were considered. The proof of the “edge-of-the-wedge” theorem by N.N. Bogolubov generated the rapid development of applications of the theory of several complex variables to axiomatic quantum field theory. During this period the “C-convex hull” and “finite covariance” theorems were proved, the Jost-Lehmann-Dyson representation was found et cetera. Recently R. Penrose has proposed a transformation connecting holomorphic solutions of the basic equations of field theory with analytic sheaf cohomologies of domains in ℝℙ3. These two directions developed, to a large extent, independently from each other, and some important results obtained in axiomatic quantum field theory still remain unknown to specialists in several complex variables and differential geometry. One of the goals of this paper is to give a unified presentation of advances in complex analysis in the future tube and related domains achieved in both of these directions.

Complex Analysis in the Future Tube

This note addresses the issue of computing the inradius and the circumradius of a convex cone in a Euclidean space. It deals also with the related problem of finding the incenter and the circumcenter of the cone. We work out various examples of convex cones arising in applications.

/pdf/inradius-and-circumradius-of-various-convex-cones-arising-in-28f3z6w3po.pdf

Inradius and Circumradius of Various Convex Cones Arising in Applications

/pdf/automorphisms-and-equivalence-of-bounded-reinhardt-domains-49hp87o2xq.pdf

Automorphisms and equivalence of bounded reinhardt domains not containing the origin

https://projecteuclid.org/download/pdf_1/euclid.pja/1195513512

Automorphisms and equivalence of tube domains with bounded base

In the study of the holomorphic automorphism group Aut( ) of a complex manifold , it seems to be natural to direct our attention not only t o the abstract group structure of Aut( ) but also to its topological group structu re equipped with the compact-open topology. In fact, a well-known theorem of H. C artan says that the topological group of the holomorphic automorphisms of a bounded omain in C has the structure of a Lie group, and this result enables us to make va rious kinds of detailed studies of bounded domains in C . On the other hand, in contrast to the case of bounded domains, the holomorphic automorphism group Aut (C × (C) ) of the unbounded domainC ×(C∗) is terribly big when +≥ 2, and cannot have the structure of a Lie group. But, by looking at topological subgroups of Au t(C × (C) ) with Lie group structures, we can find a lead to apply the Lie group theo ry t the investigation of the problems related to the structure of Aut( C × (C) ). In the present paper, we try to approach from this standpoint to the fundamental pr oblem of what complex manifold has the holomorphic automorphism group isomorphi c to Aut(C × (C) ) as topological groups. Namely, we prove the following result w i h the aid of the theory of Reinhardt domains developed in Shimizu [8], [9] (cf. Kruz hilin [6]).

/pdf/a-group-theoretic-characterization-of-the-space-obtained-by-3n69mcyj88.pdf

Satoru Shimizu

Papers

Automorphisms of bounded Reinhardt domains

Automorphisms and equivalence of bounded reinhardt domains not containing the origin

Automorphisms of bounded Reinhardt domains

Automorphisms and equivalence of tube domains with bounded base

A group-theoretic characterization of the space obtained by omitting the coordinate hyperplanes from the complex Euclidean space